Lessening the Load of Misconceptions: Design-Based Principles for Algebra Learning

References

• Adams, D. M., McLaren, B. M., Durkin, K., Mayer, R. E., Rittle-Johnson, B., Isotani, S., & van Velsen, M. (2014). Using erroneous examples to improve mathematics learning with a web-based tutoring system. Computers in Human Behavior, 36, 401–411. doi:10.1016/j.chb.2014.03.053
   Web of Science ®Google Scholar
• Adams, D. M., McLaren, B. M., Mayer, R. E., Goguadze, G., & Isotani, S. (2013) Erroneous examples as desirable difficulty. In H. C. Lane, K. Yacef, J. Mostow, & P. Pavlik (Eds.), International Conference on Artificial Intelligence in Education (AIED) (pp. 803–806). Berlin, Germany: Springer.
   Google Scholar
• Aleven, V. A. W. M. M., & Koedinger, K. R. (2002). An effective metacognitive strategy: Learning by doing and explaining with a computer-based cognitive tutor. Cognitive Science, 26, 147–179. doi:10.1207/s15516709cog2602_1
   Web of Science ®Google Scholar
• Alibali, M. W., Phillips, K. M., & Fischer, A. D. (2009). Learning new problem-solving strategies leads to changes in problem representation. Cognitive Development, 24(2), 89–101. doi:10.1016/j.cogdev.2008.12.005
   Web of Science ®Google Scholar
• Atkinson, R. K., Derry, S. J., Renkl, A., & Wortham, D. (2000). Learning from examples: Instructional principles from the worked examples research. Review of Educational Research, 70(2), 181–214. doi:10.3102/00346543070002181
   Web of Science ®Google Scholar
• Barbieri, C., & Booth, J. L. (2016). Support for struggling students in algebra: Contributions of incorrect worked examples. Learning and Individual Differences, 48, 36–44. doi:10.1016/j.lindif.2016.04.001
   Web of Science ®Google Scholar
• Berthold, K., Eysink, T. H. S., & Renkl, A. (2009). Assisting self-explanation prompts are more effective than open prompts when learning with multiple representations. Instructional Science, 37, 345–363. doi:10.1007/s11251-008-9051-z
   Web of Science ®Google Scholar
• Booth, J. L., Barbieri, C., Eyer, F., & Paré-Blagoev, E. J. (2014). Persistent and pernicious errors in algebraic problem solving. Journal of Problem Solving, 7(1), 10–23. doi:10.7771/1932-6246.1161
   Google Scholar
• Booth, J. L., Cooper, L. A., Donovan, M. S., Huyghe, A., Koedinger, K. R., & Paré-Blagoev, E. J. (2015a). Design-based research within the constraints of practice: AlgebraByExample. Journal of Education for Students Placed at Risk, 20(1–2), 79–100. doi:10.1080/10824669.2014.986674
   Web of Science ®Google Scholar
• Booth, J. L., & Koedinger, K. R. (2008). Key misconceptions in algebraic problem solving. In B. C. Love, K. McRae, & V. M. Sloutsky (Eds.), Proceedings of the 30th annual cognitive science society (pp. 571–576). Austin, TX: Cognitive Science Society.
   Google Scholar
• Booth, J. L., Koedinger, K. R., & Siegler, R. S. (2007). The effect of prior conceptual knowledge on procedural performance and learning in algebra. In D. S. McNamara & J. G. Trafton (Eds.), Proceedings of the 29th annual cognitive science society (p. 1714). Austin, TX: Cognitive Science Society.
   Google Scholar
• Booth, J. L., Lange, K. E., Koedinger, K. R., & Newton, K. J. (2013). Using example problems to improve student learning in algebra: Differentiating between correct and incorrect examples. Learning & Instruction, 25, 24–34. doi:10.1016/j.learninstruc.2012.11.002
   Web of Science ®Google Scholar
• Booth, J. L., Oyer, M. H., Paré-Blagoev, E. J., Elliot, A., Barbieri, C., Augustine, A. A., & Koedinger, K. R. (2015b). Learning algebra by example in real-world classrooms. Journal of Research on Educational Effectiveness, 8(4), 530–551. doi:10.1080/19345747.2015.1055636
   Web of Science ®Google Scholar
• Bush, S. B., & Karp, K. S. (2013). Prerequisite algebra skills and associated misconceptions of middle grade students: A review. The Journal of Mathematical Behavior, 32, 613–632. doi:10.1016/j.jmathb.2013.07.002
   Google Scholar
• Cangelosi, R., Madrid, S., Cooper, S., Olson, J., & Hartter, B. (2013). The negative sign and exponential expressions: Unveiling students’ persistent errors and misconceptions. The Journal of Mathematical Behavior, 32(1), 69–82. doi:10.1016/j.jmathb.2012.10.002
   Google Scholar
• Carroll, W. M. (1994). Using worked examples as an instructional support in the algebra classroom. Journal of Educational Psychology, 86(3), 360–367. doi:10.1037/0022-0663.86.3.360
   Web of Science ®Google Scholar
• Castle, C. (2014, September 10). University of Iowa center looks to start high school STEM academy. Press - Citizen Retrieved from http://search.proquest.com/docview/1561031112?accountid=14270
   Google Scholar
• Catrambone, R., & Yuasa, M. (2006). Acquisition of procedures: The effects of example elaborations and active learning exercises. Learning and Instruction, 16(2), 139–153. doi:10.1016/j.learninstruc.2006.02.002
   Web of Science ®Google Scholar
• Chandler, P., & Sweller, J. (1991). Cognitive load theory and the format of instruction. Cognition and Instruction, 8(4), 293–332. doi:10.1207/s1532690xci0804_2
   Web of Science ®Google Scholar
• Chase, W. G., & Simon, H. A. (1973). Perception in chess. Cognitive Psychology, 4, 55–81. doi:10.1016/0010-0285(73)90004-2
   Web of Science ®Google Scholar
• Chen, X. (2009). Students Who Study Science, Technology, Engineering, and Mathematics (STEM) in Postsecondary Education. Stats in Brief (NCES 2009-161). Washington, DC: National Center for Education Statistics.
   Google Scholar
• Chi, M. T. (2000). Self-explaining expository texts: The dual processes of generating inferences and repairing mental models. Advances in Instructional Psychology, 5, 161–238.
   Google Scholar
• Clement, J. (1982). Algebra word problem solutions: Thought processes underlying a common misconception. Journal for Research in Mathematics Education, 13, 16–30. doi:10.2307/748434
   Google Scholar
• Clement, J., Lochhead, J., & Monk, G. S. (1981). Translation difficulties in learning mathematics. The American Mathematical Monthly, 88(4), 286–290. doi:10.1080/00029890.1981.11995253
   Web of Science ®Google Scholar
• Cook, S. W., Mitchell, Z., & Goldin-Meadow, S. (2008). Gesturing makes learning last. Cognition, 106, 1047–1058. doi:10.1016/j.cognition.2007.04.010
   PubMed Web of Science ®Google Scholar
• Crooks, N. M., & Alibali, M. W. (2013). Noticing relevant problem features: Activating prior knowledge affects problem solving by guiding encoding. Frontiers in Psychology, 4. doi:10.3389/fpsyg.2013.00884
   PubMed Web of Science ®Google Scholar
• de Koning, B. B., Tabbers, H. K., Rikers, R. M. J. P., & Paas, F. (2010). Attention guidance in learning from a complex animation: Seeing is understanding? Learning and Instruction, 20, 111–122. doi:10.1016/j.learninstruc.2009.02.010
   Web of Science ®Google Scholar
• Durkin, K., & Rittle-Johnson, B. (2012). The effectiveness of using incorrect examples to support learning about decimal magnitude. Learning and Instruction, 22(3), 206–214. doi:10.1016/j.learninstruc.2011.11.001
   Web of Science ®Google Scholar
• Durkin, K., & Rittle-Johnson, B. (2015). Diagnosing misconceptions: Revealing changing decimal fraction knowledge. Learning and Instruction, 37, 21–29. doi:10.1016/j.learninstruc.2014.08.003
   Web of Science ®Google Scholar
• Fleiss, J. L. (1981). Statistical methods for rates and proportions (2nd ed.). New York, NY: John Wiley & Sons.
   Google Scholar
• Ginns, P. (2006). Integrating information: A meta-analysis of the spatial contiguity and temporal contiguity effects. Learning and Instruction, 16(6), 511–525. doi:10.1016/j.learninstruc.2006.10.001
   Web of Science ®Google Scholar
• Glaesser, Q. (1981). “Epistemologie des nombres relatifs. Recherches en Didactique des Mathématiques, 2(3), 303–346.
   Google Scholar
• Hefendehl-Hebeker, L. (1991). Negative numbers: Obstacles in their evolution from intuitive to intellectual constructs. For the Learning of Mathematics, 11(1), 26–32.
   Google Scholar
• Herriott, S. R., & Dunbar, S. R. (2009). Who takes college algebra? Primus: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 19, 74–87. doi:10.1080/10511970701573441
   Google Scholar
• Huang, X., & Reiser, R. A. (2012). The effect of instructional explanations and self-explanation prompts in worked examples on student learning and transfer. International Journal of Instructional Media, 39, 331.
   Google Scholar
• Kalchman, M., & Koedinger, K. R. (2005). Teaching and learning functions. In S. Donovan and J. Bransford (Eds.), How students learn: History, mathematics and science in the classroom (pp. 351–396). Washington, DC: National Academy Press.
   Google Scholar
• Kalyuga, S., Chandler, P., & Sweller, J. (2001). Learner experience and efficiency of instructional guidance. Educational Psychology, 21(1), 5–23. doi:10.1080/01443410124681
   Google Scholar
• Kalyuga, S., Chandler, P., Tuovinen, J., & Sweller, J. (2001). When problem solving is superior to studying worked examples. Journal of Educational Psychology, 93(3), 579. doi:10.1037/0022-0663.93.3.579
   Web of Science ®Google Scholar
• Knuth, E., Stephens, A., Blanton, M., & Gardiner, A. (2016). Build an early foundation for algebra success. Los Angeles, CA: SAGE Publications.
   Google Scholar
• Knuth, E. J., Stephens, A. C., McNeil, N. M., & Alibali, M. W. (2006). Does understanding the equal sign matter? Evidence from solving equations. Journal for Research in Mathematics Education, 37, 297–312.
   Web of Science ®Google Scholar
• Koedinger, K. R., Booth, J. L., & Klahr, D. (2013). Instructional complexity and the science to constrain it. Science, 342(6161), 935–937. doi:10.1126/science.1238056
   PubMed Web of Science ®Google Scholar
• Lee, C. Y., & Chen, M. J. (2015). Effects of worked examples using manipulatives on fifth graders’ learning performance and attitude toward mathematics. Journal of Educational Technology & Society, 18(1), 264–275.
   Web of Science ®Google Scholar
• Linchevski, L., & Williams, J. (1999). Using intuition from everyday life in “filling” the gap in children’s extension of their number concept to include the negative numbers. Educational Studies in Mathematics, 39(1–3), 131–147. doi:10.1023/A:1003726317920
   Google Scholar
• Matthews, M. S., & Farmer, J. L. (2008). Factors affecting the algebra I achievement of academically talented learners. Journal of Advanced Academics, 19(3), 472–501. doi:10.4219/jaa-2008-810
   Google Scholar
• Mayer, R. E. (2002). Multimedia learning. Psychology of Learning and Motivation, 41, 85–139.
   Web of Science ®Google Scholar
• Mayer, R. E. (2009). Multimedia learning (2nd ed.). New York, NY: Cambridge University Press. doi:10.1017/CBO9780511811678
   Google Scholar
• Mazzocco, M. M. M., Murphy, M. M., Brown, E. C., Rinne, L., & Herold, C. H. (2013). Persistent consequences of atypical early number concepts. Frontiers in Psychology, 4, 1–9. doi:10.3389/fpsyg.2013.00001
   PubMed Web of Science ®Google Scholar
• McDaniel, M. A., Frey, R. F., Fitzpatrick, S. M., & Roediger, H. L. I. I. I. (2014). Integrating cognitive science with innovative teaching in STEM disciplines. St. Louis, MO: Washington University Libraries.
   Google Scholar
• McNeil, N. M. (2008). Limitations to teaching children 2 + 2 = 4: Typical arithmetic problems can hinder learning of mathematical equivalence. Child Development, 79, 1524–1537. doi:10.1111/cdev.2008.79.issue-5
   PubMed Web of Science ®Google Scholar
• McNeil, N. M. (2014). A change–Resistance account of children’s difficulties understanding mathematical equivalence. Child Development Perspectives, 8(1), 42–47. doi:10.1111/cdep.12062
   Web of Science ®Google Scholar
• McNeil, N. M., & Alibali, M. W. (2000). Learning mathematics from procedural instruction: Externally imposed goals influence what is learned. Journal of Educational Psychology, 92, 734–744. doi:10.1037/0022-0663.92.4.734
   Web of Science ®Google Scholar
• McNeil, N. M., & Alibali, M. W. (2005). Why won’t you change your mind? Knowledge of operational patterns hinders learning and performance on equations. Child Development, 76(4), 883–899. doi:10.1111/j.1467-8624.2005.00884.x
   PubMed Web of Science ®Google Scholar
• McNeil, N. M., Fyfe, E. R., Petersen, L. A., Dunwiddie, A. E., & Brlectic-Shipley, H. (2011). Benefits of practicing 4 = 2 + 2: Nontraditional problem formats facilitate children’s understanding of mathematical equivalence. Child Development, 82, 1620–1633. doi:10.1111/j.1467-8624.2011.01622.x
   PubMed Web of Science ®Google Scholar
• Metcalfe, J. (2017). Learning from errors. Annual Review of Psychology, 68, 465–489. doi:10.1146/annurev-psych-010416-044022
   PubMed Web of Science ®Google Scholar
• Moreno, R., & Mayer, R. E. (1999). Cognitive principles of multimedia learning: The role of modality and contiguity. Journal of Educational Psychology, 91(2), 358. doi:10.1037/0022-0663.91.2.358
   Web of Science ®Google Scholar
• Mullis, I. V., Martin, M. O., Foy, P., & Arora, A. (2012). TIMSS 2011 international results in mathematics. Amsterdam, BT: International Association for the Evaluation of Educational Achievement.
   Google Scholar
• Nesher, P. (1987). Towards an instructional theory: The role of students’ misconceptions. For the Learning of Mathematics, 7, 33–40.
   Google Scholar
• Ohlsson, S. (1996). Learning from performance errors. Psychological Review, 103(2), 241–262. doi:10.1037/0033-295X.103.2.241
   Web of Science ®Google Scholar
• Ozcelik, E., Arslan-Ari, I., & Cagiltay, K. (2010). Why does signaling enhance multimedia learning? Evidence from eye movements. Computers in Human Behavior, 26(1), 110–117. doi:10.1016/j.chb.2009.09.001
   Web of Science ®Google Scholar
• Prather, R. W. (2012). Implicit learning of arithmetic regularities is facilitated by proximal contrast. PLoS one, 7(10), e48868. doi:10.1371/journal.pone.0048868
   PubMed Web of Science ®Google Scholar
• Renkl, A., Stark, R., Gruber, H., & Mandl, H. (1998). Learning from worked-out examples: The effects of example variability and elicited self-explanations. Contemporary Educational Psychology, 23(1), 90–108.
   PubMed Web of Science ®Google Scholar
• Resnick, L. B. (1983). Mathematics and science learning: A new conception. Science, 220, 477–478. doi:10.1126/science.220.4596.477
   PubMed Web of Science ®Google Scholar
• Rinne, L. F., Ye, A., & Jordan, N. C. (2017). Development of fraction comparison strategies: A latent transition analysis. Developmental Psychology, 53(4), 713. doi:10.1037/dev0000275
   PubMed Web of Science ®Google Scholar
• Rittle-Johnson, B., & Alibali, M. W. (1999). Conceptual and procedural knowledge of mathematics: Does one lead to the other? Journal of Educational Psychology, 91(1), 175–189. doi:10.1037/0022-0663.91.1.175
   Web of Science ®Google Scholar
• Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93(2), 346–362.
   Web of Science ®Google Scholar
• Roy, M., & Chi, M. T. H. (2005). Self-explanation in a multi-media context. In R. Mayer (Ed.), Cambridge handbook of multimedia learning (pp. 271–286). New York, NY: Cambridge.
   Google Scholar
• Scheiter, K., & Eitel, A. (2015). Signals foster multimedia learning by supporting integration of highlighted text and diagram elements. Learning and Instruction, 36, 11–26. doi:10.1016/j.learninstruc.2014.11.002
   Web of Science ®Google Scholar
• Schneider, B., Swanson, C. B., & Riegle-Crumb, C. (1998). Opportunities for learning: Course sequences and positional advantages. Social Psychology of Education, 2(1), 25–53. doi:10.1023/A:1009601517753
   Google Scholar
• Schwonke, R., Wittwer, J., Aleven, V., Salden, R. J. C. M., Krieg, C., & Renkl, A. (2007). Can tutored problem solving benefit from faded worked-out examples. Proceedings of EuroCogSci, 7, 59–64.
   Google Scholar
• Sidney, P. G., & Alibali, M. W. (2015). Making connections in math: Activating a prior knowledge analogue matters for learning. Journal of Cognition and Development, 16(1), 160–185. doi:10.1080/15248372.2013.792091
   Web of Science ®Google Scholar
• Siegler, R. S. (1976). Three aspects of cognitive development. Cognitive Psychology, 8(4), 481–520. doi:10.1016/0010-0285(76)90016-5
   Web of Science ®Google Scholar
• Siegler, R. S. (2002). Microgenetic studies of self-explanations. In N. Granott & J. Parziale (Eds.), Microdevelopment: Transition processes in development and learning (pp. 31–58). New York, NY: Cambridge University Press.
   Google Scholar
• Siegler, R. S., & Chen, Z. (2008). Differentiation and integration: Guiding principles for analyzing cognitive change. Developmental Science, 11(4), 433–453. doi:10.1111/j.1467-7687.2008.00689.x
   PubMed Web of Science ®Google Scholar
• Smith, J. P., III, Disessa, A. A., & Roschelle, J. (1994). Misconceptions reconceived: A constructivist analysis of knowledge in transition. The Journal of the Learning Sciences, 3(2), 115–163. doi:10.1207/s15327809jls0302_1
   Google Scholar
• Stagylidou, S., & Vosniadou, S. (2004). The development of students’ understanding of the numerical value of fractions. Learning and Instruction, 14, 503–518. doi:10.1016/j.learninstruc.2004.06.015
   Web of Science ®Google Scholar
• Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36(5), 404–411.
   Web of Science ®Google Scholar
• Sweller, J. (1988). Cognitive load during problem solving: Effects on learning. Cognitive Science, 12(2), 257–285.
   Web of Science ®Google Scholar
• Sweller, J. (2006). The worked example effect and human cognition. Learning and Instruction, 16, 165–169. doi:10.1016/j.learninstruc.2006.02.005
   Web of Science ®Google Scholar
• Sweller, J. (2012). Human cognitive architecture: Why some instructional procedures work and others do not. In K. R. Harris, S. Graham, T. Urdan, C. B. McCormick, G. M. Sinatra, & J. Sweller (Eds.), APA educational psychology handbook, Vol. 1. Theories, constructs, and critical issues (pp. 295–325). Washington, DC: American Psychological Association. doi:10.1037/13273-011
   Google Scholar
• Sweller, J., & Cooper, G. A. (1985). The use of worked examples as a substitute for problem solving in learning algebra. Cognition and Instruction, 2(1), 59–89. doi:10.1207/s1532690xci0201_3
   Google Scholar
• Vamvakoussim, X., & Vosniadou, S. (2004). Understanding the structure of the set of rational numbers: A conceptual change approach. Learning and Instruction, 14(5), 453–467. doi:10.1016/j.learninstruc.2004.06.013
   Web of Science ®Google Scholar
• Ward, M., & Sweller, J. (1990). Structuring effective worked examples. Cognition and Instruction, 7, 1–39. doi:10.1207/s1532690xci0701_1
   Web of Science ®Google Scholar
• What Works Clearinghouse. (2014). Procedures and standards handbook (Version 3.0). Washington, DC: US Department of Education.
   Google Scholar